Local Boundedness Properties for Generalized Monotone Operators

Journal of Convex Analysis Volume 19 (2012), No. 1, 49–61

Mohammad Hossein Alizadeh Department of Product and Systems Design Engineering, University of the Aegean, 84100 Hermoupolis, Syros, Greece [email protected] Nicolas Hadjisavvas∗ Department of Product and Systems Design Engineering, University of the Aegean, 84100 Hermoupolis, Syros, Greece [email protected] Mehdi Roohi Department of Mathematics, Faculty of Basic Sciences, University of Mazandaran, Babolsar, 47416 – 1468, Iran [email protected]

Received: June 17, 2010 Revised manuscript received: January 23, 2011

We show that a well-known property of monotone operators, namely local boundedness in the interior of their domain, remains valid for the larger class of premonotone maps. This generalizes a similar result by Iusem, Kassay and Sosa (J. Convex Analysis 16 (2009) 807–826) to Banach spaces.

Keywords: Monotone operator, generalized monotone operator, locally bounded operator

2010 Mathematics Subject Classiﬁcation: 47H05, 49J53, 47H04

1. Introduction In recent years, operators which have some kind of generalized monotonocity property have received a lot of attention (see for example [7] and the references therein). Many papers considering generalized monotonicity were devoted to the investigation of its relation to generalized convexity; others studied the existence of solutions of generalized monotone variational inequalities and, in some cases, derived algorithms for finding such solutions. Monotone operators are known to have many very interesting properties. For instance, it is known that a monotone operator T defined on a Banach space is locally bounded in the interior of its domain. Actually by the Libor Veselýtheorem, whenever T is maximal monotone and D(T ) is convex, interior points of the domain D(T ) are the only points of D(T ) where T is locally bounded. So the question nat- ∗Corresponding author

ISSN 0944-6532 / $ 2.50 c Heldermann Verlag 50 M. H. Alizadeh, N. Hadjisavvas, M. Roohi / Local Boundedness Properties ... urally arises: are these properties shared by other operators which satisfy a more relaxed kind of monotonicity? In a recent paper, Iusem, Kassay and Sosa [8] introduced the class of the so-called premonotone operators. This class includes monotone operators, but contains many more: for example, if T is monotone and R is globally bounded, then T + R is premonotone. In fact, it includes ε-monotone operators which are related to the very useful ε-subdifferentials [9, 10]. In [8] it is shown that, in a finite dimensional space, premonotone operators are locally bounded in the interior of their domain. The proof was based on the finite dimensionality of the space. In this paper we will show that these results remain valid in infinite dimensional Banach spaces. We also show that some properties of monotone operators remain valid in a much more general context. More precisely, after some preliminary definitions and results in Section 2, we introduce in Section 3 the notion of a premonotone bifunction. We will show that such bifunctions are locally bounded in the interior of their domain and we will deduce local boundedness of premonotone operators. We will also show a generalization of the Libor Veselýtheorem. Let us fix some notation and recall some definitions. In what follows, X will be a ∗ Banach space. Given a multivalued operator T : X → 2X , we will set D(T ) = {x ∈ X : T (x) =6 ∅} and gr(T ) = {(x, x∗) ∈ X ×X∗ : x∗ ∈ T (x)} to be its domain and its graph, respectively. T is called norm×weak∗ closed (resp., sequentially norm×weak∗ closed) if gr(T ) is closed (resp., sequentially closed) in the norm×weak∗ topology ∗ ∗ of X × X . Given x0 ∈ X, T is called sequentially norm×weak closed at x0 if ∗ ∗ ∗ for every sequence (xn, xn) ∈ gr(T ) such that xn → x0 and xn weak -converges to ∗ ∗ ∗ ∗ ∗ some x0 ∈ X , one has x0 ∈ T (x0).T is called monotone if hx − y , x − yi ≥ 0 for all (x, x∗), (y, y∗) ∈ gr(T ). It is called maximal monotone if it is monotone, and for each monotone operator S such that gr(T ) ⊆ gr(S), one has T = S. The operator T is called locally bounded at x0 ∈ X if there exists some neighborhood V of x0 such that the set T (V ) := x∈V T (x) is bounded.

Given a convex set C ⊆ XSand x ∈ C we will denote by NC (x) the normal cone of C at x: ∗ ∗ ∗ NC (x) = {x ∈ X : ∀y ∈ C, hx , y − xi ≤ 0}.

2. σ-Monotone operators Most definitions and many of the results of this section are essentially due to [8], the main difference being that in [8] one considers premonotone operators in Rn, without specifying a given σ. X∗ Definition 2.1. (i) Given an operator T : X → 2 and a map σ : D(T ) → R+, T is said to be σ-monotone if for every x, y ∈ D (T ) , x∗ ∈ T (x) and y∗ ∈ T (y),

hx∗ − y∗, x − yi ≥ − min{σ(x), σ(y)}kx − yk. (1)

(ii) An operator T is called premonotone if it is σ-monotone for some σ : D(T ) → R+. M. H. Alizadeh, N. Hadjisavvas, M. Roohi / Local Boundedness Properties ... 51

(iii) A σ-monotone operator T is called maximal σ-monotone, if for every operator T ′ which is σ′-monotone with gr(T ) ⊆ gr(T ′) and σ′ an extension of σ, one has T = T ′.

The notion of premonotone operators for the ﬁnite-dimensional case is introduced in [8]. The same paper also contains examples of maximal σ-monotone operators. ∗ Remark 2.2. (i) It should be noticed that T : X → 2X is σ-monotone if and only if

∀x, y ∈ D (T ) , x∗ ∈ T (x) , y∗ ∈ T (y) , hx∗ − y∗, x − yi ≥ −σ(y)kx − yk. (2)

(ii) If σ(y) = 2ε ≥ 0 for each y ∈ D(T ), then T is called ε-monotone [10]. (iii) Definition 2.1 does not allow negative values for σ, since this can only happen in very special cases. For instance, if T satisfies (1) and its domain contains any line segment [x0, y0] := {(1−t)x0 +ty0 : t ∈ [0, 1]}, then the set of points x ∈ [x0, y0] where σ(x) < 0 is at most countable. Indeed, if this is not the case, then there exists ∗ ε > 0 such that σ(x) < −ε for infinitely many x ∈ [x0, y0]. Choose x0 ∈ T (x0), ∗ N y0 ∈ T (y0). Given n ∈ , choose xk = x0 + tk(y0 − x0), k = 1, . . . n − 1, such that ∗ 0 < t1 < ··· < tn−1 < 1 and σ(xk) < −ε. Then choose xk ∈ T (xk). Set xn = y0 and ∗ ∗ xn = y0. Relation (2) gives for all k = 0, 1, . . . n − 1:

∗ ∗ xk+1 − xk, xk+1 − xk ≥ ε kxk+1 − xkk ⇒ ∗ ∗ xk+1 − xk, y0 − x0 ≥ ε ky0 − x0k .

∗ ∗ Adding these inequalities for k = 0, 1, . . . n−1 yields hy0 − x0, y0 − x0i ≥ nεky0 − x0k. This should hold for each n ∈ N, which is impossible. (iv) The notion of premonotonicity is not suited to linear operators, since every σ-monotone linear operator T : X → X∗ is in fact monotone. Indeed, in this case putting y = 0 in (2) we ﬁnd

∀x ∈ X, hT x, xi ≥ −σ (0) ||x||. (3)

Replacing x by x − y we deduce that

∀x, y ∈ X, hT x − T y, x − yi ≥ −σ (0) ||x − y||.

Thus T is ε-monotone with ε = σ (0) /2. Then Proposition 3.2 in [10] shows that T is monotone. (v) A σ-monotone operator is maximal σ-monotone if and only if, for every operator T ′ which is σ′-monotone with gr(T ) ⊆ gr(T ′) and σ′(x) ≤ σ(x) for all x ∈ D(T ), one has T = T ′.

The following proposition is an easy consequence of Zorn’s Lemma, as for monotone operators. 52 M. H. Alizadeh, N. Hadjisavvas, M. Roohi / Local Boundedness Properties ...

Proposition 2.3. Every σ-monotone operator has a maximal σ-monotone extension.

Deﬁnition 2.4. Let A be a subset of X. Given a mapping σ : A → R+, two pairs (x, x∗), (y, y∗) ∈ A × X∗ are σ-monotonically related if hx∗ − y∗, x − yi ≥ − min{σ(x), σ(y)}kx − yk.

The proof of the following proposition is obvious. ∗ Proposition 2.5. The σ-monotone operator T : X → 2X is maximal σ-monotone ∗ ∗ ′ if and only if, for every point (x0, x0) ∈ X × X and every extension σ of σ to ∗ ′ ∗ D(T )∪{x0} such that (x0, x0) is σ -monotonically related to all pairs (y, y ) ∈ gr(T ), ∗ we have (x0, x0) ∈ gr(T ).

X∗ Given an operator T : X → 2 , we deﬁne the function σT : D(T ) → R+ ∪ {∞} by ∗ ∗ ∗ ∗ σT (y) = inf{a ∈ R+ : hx − y , x − yi ≥ −a kx − yk , ∀(x, x ) ∈ gr(T ), y ∈ T (y)}.

Note that if the operator T is premonotone, then

σT = inf{σ : T is σ-monotone} and thus σT is ﬁnite, and T is σT -monotone. Also in this case, it is obvious that hx∗ − y∗, y − xi σ (y) = max sup{ : x ∈ X\{y}, x∗ ∈ T (x), y∗ ∈ T (y)}, 0 (4) T ky − xk (see also [8]). The following result is due to [8]. Proposition 2.6. Let an operator T be given.

(i) σT is ﬁnite and T is σT -monotone, if and only if T is σ-monotone for some σ. (ii) σT is ﬁnite and T is maximal σT -monotone, if and only if T is maximal σ-monotone for some σ.

Proof. We have only to prove that whenever T is maximal σ-monotone for some X∗ ′ σ, then it is maximal σT -monotone. Assume that S : X → 2 is σ -monotone with ′ ′ gr(T ) ⊆ gr(S) and σ an extension of σT . Since σ = σT ≤ σ on D(T ), by Remark 2.2(v) we get that S = T . Hence, T is maximal σT -monotone. Proposition 2.7. Every maximal σ-monotone operator T is convex-valued and ∗ weak closed-valued. Moreover, if σ is deﬁned and usc at some point x0 ∈ D(T ), ∗ then T is sequentially norm×weak closed at x0.

X∗ ∗ ∗ Proof. Let T : X → 2 be a maximal σ-monotone operator and (x, x1), (x, x2) ∈ gr(T ), λ ∈ [0, 1]. Then for each (y, y∗) ∈ gr(T ), ∗ ∗ ∗ hλx1 + (1 − λ)x2 − y , x − yi ∗ ∗ ∗ ∗ = λhx1 − y , x − yi + (1 − λ)hx2 − y , x − yi ≥ − λ min{σ(x), σ(y)}kx − yk − (1 − λ) min{σ(x), σ(y)}kx − yk = − min{σ(x), σ(y)}kx − yk. M. H. Alizadeh, N. Hadjisavvas, M. Roohi / Local Boundedness Properties ... 53

∗ ∗ ∗ That is, (x, λx1 + (1 − λ)x2) is σ-monotonically related with all (y, y ) ∈ gr(T ). ∗ ∗ Now, it follows from Proposition 2.5 that (x, λx1 +(1−λ)x2) ∈ gr(T ) which implies that T (x) is convex. Likewise, one can show that T (x) is weak∗ closed. ∗ We now show sequential closedness: suppose that (xn, xn) is a sequence in gr(T ) ∗ ∗ w ∗ such that xn → x0 and xn → x0. Assume that σ is usc at x0. It follows from the σ-monotonicity of T that for each (y, y∗) ∈ gr(T ) we have

∗ ∗ hxn − y , xn − yi ≥ − min{σ(xn), σ(y)}kxn − yk. By taking limits in the above inequality and using the upper semicontinuity of σ at ∗ x0 and the fact that {xn} is a bounded sequence, we get

∗ ∗ hx0 − y , x0 − yi ≥ − min{σ(x0), σ(y)}kx0 − yk

∗ ∗ which implies that (x0, x0) is σ-monotonically related with all (y, y ) ∈ gr(T ). By ∗ using Proposition 2.5 we deduce that (x0, x0) ∈ gr(T ).

We note that, as for monotone operators, in general gr(T ) is only sequentially norm×weak∗ closed, not norm×weak∗ closed [5]. However, we will see in the next section that maximal σ-monotone operators are actually usc in the interior of their domains. The assumption of upper semicontinuity of σ cannot be omitted from Proposition 2.7, as the following example shows. This is also an example of a premonotone operator which is not ε-monotone. Note that for T : R → R we have

σT (y) = max sup {T (x) − T (y)} , sup {T (y) − T (x)} . (5) x≤y x≥y Example 2.8. We deﬁne the functions ϕ, σ : R → R by

x sin2 x if x ≥ 0, ϕ(x) = (0 if x < 0, and σ(x) = max ϕ(x), max ϕ(z) − ϕ(x) . z≤x We show that ϕ is σ-monotone, i.e., for all x, y ∈ R the following inequality holds:

(ϕ(x) − ϕ(y)) (x − y) ≥ − min{σ(x), σ(y)} |x − y| .

We may assume without loss of generality that x ≤ y, so we have to prove that ϕ(x) − ϕ(y) ≤ min{σ(x), σ(y)}. Indeed,

ϕ(x) − ϕ(y) ≤ ϕ(x) ≤ σ(x) and ϕ(x) − ϕ(y) ≤ max ϕ(z) − ϕ(y) ≤ σ(y) z≤y 54 M. H. Alizadeh, N. Hadjisavvas, M. Roohi / Local Boundedness Properties ... so ϕ is σ-monotone. Note that ϕ is not ε-monotone since (ϕ(x) − ϕ(y)) sgn (x − y) is not bounded from below (take y = 2kπ + π/2, x = 2kπ + π for large k ∈ N).

We now change ϕ and σ at one point: deﬁne T, σ1 : R → R by π π ϕ(x) if x =6 , σ(x) if x =6 , 2 2 T (x) = π π and σ1(x) = π π  if x = ,  if x = .  4 2  4 2   One can readily show that T is σ1-monotone.

Now let T be a maximal σ1-monotone extension of T . Its graph is not closed; indeed (π/2, π/2) belongs to the closure of gr(T ). However, it does not belong to gr(T ) sincee it is not σ1-monotonically related to (π, 0) ∈ gr(T ): since σ1(π) = maxz≤π ϕ(z) ≥ ϕ(π/2) = π/2, one has e e e π π π π π − 0 sgn − π = − < − = − min σ , σ (π) . 2 2 2 4 1 2 1 n o 3. Local boundedness and related properties Some properties of σ-monotone operators can be more easily investigated through the use of σ-monotone bifunctions that we now introduce. Let X be a Banach space, C a nonempty subset of X and σ : C → R+ be a map. A bifunction F : C ×C → R will be called σ-monotone if

∀x, y ∈ C,F (x, y) + F (y, x) ≤ min{σ (x) , σ(y)}kx − yk. (6)

Equivalently, F is σ-monotone if

∀x, y ∈ C,F (x, y) + F (y, x) ≤ σ(y)kx − yk. (7)

This notion is a generalization of the notion of monotone bifunction introduced in [4], where σ is identically zero. Given any bifunction F : C × C → R, we deﬁne as in [1, 3] the operator AF : X → ∗ 2X by

{x∗ ∈ X∗ : ∀y ∈ C,F (x, y) ≥ hx∗, y − xi} if x ∈ C, AF (x) = ( ∅ if x∈ / C.

Note that in case F (x, x) = 0 for all x ∈ C, one has AF (x) = ∂F (x, ·)(x) (the subdiﬀerential of the function F (x, ·) at x). Proposition 3.1. For a σ-monotone bifunction F , AF is σ-monotone.

Proof. Let x∗ ∈ AF (x) and y∗ ∈ AF (y). By the deﬁnition of AF ,

F (x, y) ≥ hx∗, y − xi M. H. Alizadeh, N. Hadjisavvas, M. Roohi / Local Boundedness Properties ... 55 and F (y, x) ≥ hy∗, x − yi. From these inequalities we obtain

hx∗ − y∗, x − yi ≥ −F (x, y) − F (y, x) ≥ − min{σ (x) , σ(y)}kx − yk.

Deﬁnition 3.2. A σ-monotone bifunction F is called maximal σ-monotone if AF is maximal σ-monotone.

X∗ For a given operator T : X → 2 , as in [6] we deﬁne GT : D(T ) × D(T ) → R ∗ ∪ {+∞} by GT (x, y) = supx∗∈T (x)hx , y − xi. For each x ∈ D(T ), GT (x, ·) is lsc and convex, and GT (x, x) = 0. The following result shows that GT is actually real valued whenever T is σ-monotone, and establishes some relations between σ- monotonicity of GT and T . Proposition 3.3. Let T be an operator. Then the following statements are true.

(i) If T is σ-monotone, then GT is a real-valued, σ-monotone bifunction. (ii) If T is maximal σ-monotone, then GT is a maximal σ-monotone bifunction and AGT = T . (iii) Suppose that T is σ-monotone with closed convex values and D(T ) = X. If GT is maximal σ-monotone, then T is maximal σ-monotone.

∗ Proof. (i) Let T : X → 2X be σ-monotone. Given x, y ∈ D (T ), for every x∗ ∈ T (x) and y∗ ∈ T (y), we have

hx∗ − y∗, x − yi ≥ −σ(y)kx − yk.

Thus hy∗, x − yi + hx∗, y − xi ≤ σ(y)kx − yk. This implies that

sup hy∗, x − yi + sup hx∗, y − xi ≤ σ(y)kx − yk. y∗∈T (y) x∗∈T (x)

Form here we conclude that

∀x, y ∈ D (T ) ,GT (x, y) + GT (y, x) ≤ σ(y)kx − yk.

Consequently, GT (x, y) ∈ R for all x, y ∈ D (T ) and GT is a σ-monotone bifunction. (ii) Let (x, z∗) ∈ gr(T ). For every y ∈ C we have

∗ ∗ GT (x, y) = sup hx , y − xi ≥ hz , y − xi. x∗∈T (x)

This means that z∗ ∈ AGT (x); i.e., T (x) ⊆ AGT (x). It follows from Proposition 3.1 and part (i) that AGT is σ-monotone. Since T is maximal σ-monotone, we conclude that T = AGT . 56 M. H. Alizadeh, N. Hadjisavvas, M. Roohi / Local Boundedness Properties ...

GT (iii) Since GT is maximal σ-monotone by assumption, A is maximal σ-monotone. Let x ∈ X and z∗ ∈ AGT (x). Then

∗ ∗ GT (x, y) = sup hx , y − xi ≥ hz , y − xi. x∗∈T (x)

Now, the separation theorem implies that z∗ ∈ T (x). Thus, gr(AGT ) ⊆ gr(T ). This implies that T = AGT and T is maximal σ-monotone.

Remark 3.4. Given a maximal σ-monotone bifunction F , according to Proposition F 3.3, we can construct A and the σ-monotone bifunction G := GAF . One has G(x, y) ≤ F (x, y) for all x, y ∈ D(AF ). It follows from Proposition 3.3 that AF = AG. However, Example 2.5 of [6] implies that the correspondence F 7→ AF is not one to one, even for the monotone case σ ≡ 0.

We now generalize a deﬁnition from [6]. Deﬁnition 3.5. A bifunction F : C × C → R is called:

(i) Locally bounded at (x0, y0) ∈ X × X if there exist an open neighborhood V of x0, an open neighborhood W of y0 and M ∈ R such that F (x, y) ≤ M for all (x, y) ∈ (V × W ) ∩ (C × C). (ii) Locally bounded on K × L ⊆ X × X, if it is locally bounded at each (x, y) ∈ K × L.

(iii) Locally bounded at x0 ∈ X if it is locally bounded at (x0, x0), i.e., there exist an open neighborhood V of x0 and M ∈ R such that F (x, y) ≤ M for all x,y ∈ V ∩ C. (iv) Locally bounded on K ⊆ X, if it is locally bounded at each x ∈ K.

If a bifunction (not necessarily σ-monotone) F : C × C → R is locally bounded at F x0 ∈ int C, then A is locally bounded at x0 [2]. Consequently, if T is an operator such that GT is locally bounded at x0 ∈ int D(T ), then T is locally bounded at x0 since T (x) ⊆ AGT (x) for all x ∈ X. This will be the main instrument for showing local boundedness of operators. We will show that σ-monotone bifunctions are locally bounded in the interior of their domain, under mild assumptions. In case X = Rn we can give a constructive proof. Proposition 3.6. Let X = Rn and C ⊆ Rn. Assume that F : C × C → R is σ-monotone and F (x, ·) is lsc and quasiconvex for every x ∈ C. Then F is locally bounded at every point of int C × int C.

Proof. Let (x0, y0) ∈ int C × int C. Since the space is finite-dimensional, we can find z1, z2, . . . , zm ∈ C such that V := co{z1, z2, . . . , zm} ⊆ C is a neighborhood of y0. Let U ⊆ C be a compact neighborhood of x0 in C. Set Mk = minx∈U F (zk, x); the minimum exists since F (zk, ·) is lsc. For every x ∈ U, y ∈ V we find, using M. H. Alizadeh, N. Hadjisavvas, M. Roohi / Local Boundedness Properties ... 57 quasiconvexity of F (x, ·) and σ-monotonicity of F :

F (x, y) ≤ max F (x, zk) 1≤k≤m

≤ max {σ(zk) kx − zkk − F (zk, x)} 1≤k≤m

≤ max σ(zk) sup kz − wk + max (−Mk). 1≤k≤m z∈U,w∈V 1≤k≤m

Since U and V are both bounded, supz∈U,w∈V kz − wk is ﬁnite. We are done.

For the general case of a Banach space X, we need the following lemma from [2], whose proof we include for the sake of completeness. Lemma 3.7 ([2]). Let X be a Banach space and f : X → R ∪ {+∞} be lsc and quasiconvex. If x0 ∈ int dom(f), then f is bounded from above on a neighborhood of x0.

Proof. Let ε > 0 be such that B(x0, ε) ⊆ dom(f). Set Sn = {x ∈ B(x0, ε): f(x) ≤ n}. Then Sn are convex and closed and n∈N Sn = B(x0, ε). By Baire’s theorem, there exists n ∈ N such that int Sn =6 ∅. Take any x1 ∈ int Sn and any x2 =6 x0 S such that x2 ∈ B(x0, ε) and x0 ∈ co{x1, x2}. Choose n1 > max{n, f(x2)}. Then x1 ∈ int Sn1 , x2 ∈ Sn1 hence x0 ∈ int Sn1 so f is bounded by n1 at a neighborhood of x0. Theorem 3.8. Suppose X is a Banach space, C is a subset of X and F : C × C → R is a σ-monotone bifunction such that for every x ∈ C, F (x, ·) is lsc and quasiconvex. Further, suppose that for some x0 ∈ C and y0 ∈ int C there exists ε > 0 such that B(y0, ε) ⊆ C and for each y ∈ B(y0, ε), F (y, ·) is bounded from below on B(x0, ε) ∩ C (note that this bound may depend on y). Then F is locally bounded at (x0, y0).

Proof. Let ε > 0 be as in the assumption. Deﬁne g : B(y0, ε) → R ∪ {+∞} by

g(y) := sup{F (x, y): x ∈ B(x0, ε) ∩ C}.

For every y ∈ B(y0, ε) and x ∈ B(x0, ε) ∩ C, σ-monotonicity of F implies

F (x, y) ≤ min{σ(x), σ(y)}kx − yk − F (y, x) ≤ σ(y)(ε + ky − x0k) − My where My is a lower bound of F (y, ·) on B(x0, ε) ∩ C. Therefore, g is real-valued. On the other hand, g is lsc and quasiconvex and also y0 ∈ int dom(g). By Lemma 3.7, there exists δ < ε and M ∈ R such that g(y) ≤ M for all y ∈ B(y0, δ). Then by the deﬁnition of g we get F (x, y) ≤ M for all y ∈ B(y0, δ) and x ∈ B(x0, δ) ∩ C; i.e., F is locally bounded at (x0, y0).

The condition “F (y, ·) is bounded from below on B(x0, ε)∩C can be easily removed by imposing some usual assumptions on the bifunction F or the space X, as shown in the following two results. 58 M. H. Alizadeh, N. Hadjisavvas, M. Roohi / Local Boundedness Properties ...

Corollary 3.9. Suppose X is a reﬂexive Banach space, C is a subset of X and F : C × C → R is a σ-monotone bifunction such that for every x ∈ C, F (x, ·) is lsc and quasiconvex. Then F is locally bounded at every point of int C × int C. If in addition C is weakly closed, then F is locally bounded on C × int C.

Proof. Let x0 ∈ int C. Choose ε > 0 such that B(x0, ε) ⊆ C. By assumption F (x, ·) is lsc and quasiconvex, so it is weakly lsc. For every y ∈ C, F (y, ·) attains its minimum on the weakly compact set B(x0, ε) and so F (y, ·) is bounded from below on B(x0, ε). Therefore, all conditions of Theorem 3.8 are satisﬁed. Thus F is locally bounded at every point of int C × int C.

If in addition C is weakly closed, then for any x0 ∈ C and ε > 0, B(x0, ε) ∩ C is weakly compact and we can repeat the previous argument.

Corollary 3.10. Suppose X is a Banach space, C is a subset of X and F : C×C → R is a σ-monotone bifunction such that for every x ∈ C, F (x, ·) is lsc and convex. Then F is locally bounded at any point of C × int C.

Proof. Let x0 ∈ C and y0 ∈ int C. Choose ε > 0 such that B(y0, ε) ⊆ C. For ∗ every y ∈ B(y0, ε), the subdiﬀerential of ∂F (y, ·) is nonempty at y. Choose y ∈ ∂F (y, ·)(y). Then for every x ∈ B(x0, ε) ∩ C one has

∗ ∗ ∗ F (y, x) − F (y, y) ≥ hy , x − yi ≥ − ky k kx − yk ≥ − ky k (ε + kx0 − yk).

Thus F (y, ·) is bounded from below on B(x0, ε) ∩ C. By Theorem 3.8, F is locally bounded at (x0, y0).

We immediately obtain a generalization of Proposition 3.5 in [8] to general Banach spaces: ∗ Corollary 3.11. Suppose that X is a Banach space and T : X → 2X is a premonotone operator. Then T is locally bounded at every point of int D(T ).

Proof. Apply Corollary 3.10 to GT .

Corollary 3.12 (Rockafellar). Every set valued monotone operator T from X to X∗ is locally bounded on int D(T ).

For maximal σ-monotone operators, there is a kind of converse to Corollary 3.11, generalizing the Libor Vesel´ytheorem [11, Theorem 1.14]. We ﬁrst show: Lemma 3.13. If T is maximal σ-monotone, then for all x ∈ D (T ) one has T (x)+ ND(T ) (x) ⊆ T (x).

∗ Proof. Take w ∈ ND(T ) (z) and deﬁne

T (x) if x =6 z, T (x) = 1 R ∗ (T (x) + +w if x = z. M. H. Alizadeh, N. Hadjisavvas, M. Roohi / Local Boundedness Properties ... 59

∗ ∗ Then T (x) ⊆ T1 (x) for all x ∈ D(T ). For z ∈ T (z) , y ∈ T (y) and λ > 0, hz∗ + λw∗ − y∗, z − yi = hz∗ − y∗, z − yi + λhw∗, z − yi ≥ − min {σ (z) , σ (y)} ||z − y||.

Thus T1 is σ-monotone. By the maximality of T we get T = T1, which completes the proof. Theorem 3.14. Suppose that T is maximal σ-monotone and σ is deﬁned and usc on D (T ). Let x0 ∈ D (T ). If T is locally bounded at x0, then x0 ∈ D (T ). If in addition D (T ) is convex, then x0 ∈ int D(T ).

Proof. Since T is locally bounded at x0, there exists an open neighborhood U of x0 such that T (U) is bounded. Choose a sequence {xn} ⊆ D (T ) ∩ U such that ∗ xn → x0 and choose xn ∈ T (xn). It follows from Alaoglou’s theorem that there ∗ ∗ ∗ ∗ ∗ ∗ w ∗ exist a subnet {(xα, xα)} of {(xn, xn)} and x0 ∈ X such that xα → x0. Since the ∗ ∗ ∗ net {xα} is in the bounded set T (U), we have hxα, xαi → hx0, x0i. Therefore for all (y, y∗) ∈ gr (T ), by upper semicontinuity of σ,

∗ ∗ ∗ ∗ hx − y , x0 − yi = limhxα − y , xα − yi 0 α

≥ − lim sup min {σ (xα) , σ (y)} ||xα − y|| α

≥ − min {σ (x0) , σ (y)} ||x0 − y||.

∗ ∗ ∗ Thus (x0, x0) is σ-monotonically related with all (y, y ) ∈ gr (T ). So x0 ∈ T (x0) and x0 ∈ D(T ). Now let D (T ) be convex. We will show that U ⊆ int D (T ). Indeed, if not, then U contains a boundary point of D (T ). By the Bishop-Phelps theorem it will also contain a support point of D (T ), i.e., there exist z ∈ U ∩ D (T ) and 0 =6 w∗ ∈ X∗ such that hw∗, zi = sup{hw∗, yi : y ∈ D (T )}. We know that T is locally bounded ∗ at z, hence z ∈ D(T ). On the other hand, w ∈ ND(T )(z), thus the cone ND(T )(z) is not equal to {0}. Then Lemma 3.13 shows that T (z) cannot be bounded, a contradiction. Thus U ⊆ int D (T ). Since T is locally bounded on U, we obtain U ⊆ D(T ), hence x0 ∈ int D(T ).

We now deduce some properties related to local boundedness. ∗ Proposition 3.15. Suppose T : X → 2X is maximal σ-monotone and σ is usc. Then (i) The operator T is usc in int D(T ) from the norm topology in X to the weak∗ topology in X∗; (ii) If X is ﬁnite-dimensional, then for every y ∈ int D(T ), σT (y) is given by the following formula:

hx∗ − y∗, y − xi σ (y) = sup : x =6 y, (x, x∗) ∈ gr(T ), y∗ ∈ T (y) . (8) T ky − xk 60 M. H. Alizadeh, N. Hadjisavvas, M. Roohi / Local Boundedness Properties ...

Proof. Fix y ∈ int D(T ). To show upper semicontinuity at y, it is suﬃcient to show ∗ ∗ that for any net {(yα, yα)} in gr(T ) such that yα → y in X, there exists a weak ∗ cluster point of {yα} in T (y). Since T is locally bounded at y we may assume that ∗ ∗ ∗ w ∗ both {yα} and {yα} are bounded and, by selecting a subnet if necessary, yα → y . ∗ Since {yα} is bounded, we have

∗ ∗ hyα, yαi → hy , yi .

As in the proof of Proposition 2.7 we deduce that y∗ ∈ T (y).

To show part (ii), choose any sequence {xn}n∈N ⊆ D(T ) converging to y with ∗ ∗ y =6 xn, and let xn ∈ T (xn). Then the sequence {xn} is bounded. By selecting ∗ a subsequence if necessary, we may assume that xn converges in norm to some z∗ ∈ T (y). Since

hx∗ − y∗, y − xi hx∗ − z∗, y − x i sup : x =6 y, (x, x∗) ∈ gr(T ), y∗ ∈ T (y) ≥ n n ky − xk ky − x k n ∗ ∗ ≥ − kxn − z k → 0, relation (8) follows from relation (4).

Next we show that under appropriate conditions, a σ-monotone bifunction is not only locally bounded, but also bounded by a small number in a neighborhood of any interior point. This is a consequence of the following more general result. Proposition 3.16. Suppose that F : C × C → R is a σ-monotone bifunction such that F (x, x) = 0 for all x ∈ C. Assume that F (x, ·) is lsc and convex for each x ∈ C and σ is usc. If x0 ∈ int C, then there exist an open neighborhood V of x0 and K ∈ R such that F (y, x) ≤ K kx − yk for all x ∈ V and y ∈ C.

Proof. From F (x, x) = 0 for all x ∈ C, we infer that AF (x) = ∂F (x, ·)(x). Since F (x, ·) is lsc and convex, the subdiﬀerential of F (x, ·) at each x ∈ int C is nonempty- valued. Thus int C ⊆ D(AF ), so the σ-monotone operator AF is locally bounded at x0. Therefore, there exist an open neighborhood V1 ⊆ C of x0 and K1 ∈ R such ∗ ∗ F that kx k ≤ K1 for all x ∈ A (x), x ∈ V1. Since σ is usc at x0, it is bounded from above by a number K2 on a neighborhood V2 of x0. Then for each y ∈ C and ∗ F x ∈ V := V1 ∩ V2, if we choose x ∈ A (x) we get

F (y, x) ≤ − F (x, y) + σ(x) ky − xk ∗ ≤ − hx , y − xi + K2 ky − xk ≤ (K1 + K2) ky − xk .

Acknowledgements. The authors are grateful to the referee for his very careful reading of the paper which led to some crucial improvements. M. H. Alizadeh, N. Hadjisavvas, M. Roohi / Local Boundedness Properties ... 61

References [1] M. Ait Mansour, Z. Chbani, H. Riahi: Recession bifunction and solvability of nonco- ercive equilibrium problems, Commun. Appl. Anal. 7 (2003) 369–377. [2] M. H. Alizadeh, N. Hadjisavvas: Local boundedness of monotone bifunctions, J. Glob. Optim., to appear, DOI 10.1007/s10898-011-9677-2. [3] K. Aoyama, Y. Kimura, W. Takahashi: Maximal monotone operators and maximal monotone functions for equilibrium problems, J. Convex Analysis 15 (2008) 395–409. [4] E. Blum, W. Oettli: From optimization and variational inequalities to equilibrium problems, Math. Stud. 63 (1994) 123–145. [5] J. Borwein, S. Fitzpatrick, R. Girgensohn: Subdifferentials whose graphs are not norm ×weak∗ closed, Canad. Math. Bull. 46 (2003) 538–545. [6] N. Hadjisavvas, H. Khatibzadeh: Maximal monotonicity of bifunctions, Optimization 59 (2010) 147-160. [7] N. Hadjisavvas, S. Komlósi,S. Schaible: Handbook of Generalized Convexity and Generalized Monotonicity, Springer, New York (2005). [8] A. N. Iusem, G. Kassay, W. Sosa: An existence result for equilibrium problems with some surjectivity consequences, J. Convex Analysis 16 (2009) 807–826. [9] A. Jofré,D. T. Luc, M. Théra: ε-subdifferential and ε-monotonicity, Nonlinear Anal. 33 (1998) 71–90. [10] D. T. Luc, H. V. Ngai, M. Théra:On ε-monotonicity and ε-convexity, in: A. Ioffe et al. (eds.), Calculus of Variations and Differential Equations (Haifa, 1998), Res. Notes Math. Ser. 410, Chapman & Hall, Boca Raton (1999) 82–100. [11] R. Phelps: Lectures on maximal monotone operators, Extr. Math. 12 (1997) 193–230.